Will you need a mortgage on top of the proceeds of the sale? If so, getting approval in principle in advance will also be important as EAs will ask you to prove affordability.
Unfortunately I don't think anyone will give me a mortgage. I went self employed a few months before covid hit, and am (partially) in an industry that was hard hit by covid. In essence I don't have long term accounts and as I understand it mortgage lenders would be very nervous about lending to anyone whose income has been hit by covid.
Thankfully I'm in a cheaper part of the country, don't need such a large house, or to live in a fashionable area, so I should be able to buy somewhere outright with my portion of the sale (or at the very least to be able to top it up with the deposit I'd been saving for).
If probate has been granted before the house goes on the market, and it's likely to sell fast (sounds like it) then as long as you have a good, proceedable buyer, and your onward purchase doesn't have a big/difficult chain, then there's no reason to think you couldn't be in by September. There are a lot of dependencies there though and you'll have to be strategic. Who is driving the probate process? Is the executor on top of it? Do you have a good solicitor? Probate is back to a more normal time now but there are lots of things to do - the whole estate needs to be managed, not just the house (i.e. all accounts found and closed, documentation provided etc.) so with Christmas and potentially another Covid winter this could take longer than expected. You may have to chase it yourself to keep it moving.
The idea is to get probate granted before the house goes on the market. I've heard of there being difficulties with getting builders and materials, but there are few doer-uppers in the local area (I've checked RM!) so someone looking for a project has limited choice. Fully renovated it would be worth about £1.1m so ought to attract at least one decent buyer. Hopefully.
The executors are my DM and DAunt - sadly but helpfully DAunt has been an executor several times in the last few years, so is familiar with the form filling, and is quite an efficient person anyway. DM is more local and is dealing with the more practical side like clearing the house (a mammoth job!). No solicitor has been mentioned, I think all things IHT are being done on a DIY basis. I know they have been tackling the accounts as well as the house. Fingers crossed there won't be another covid lockdown that prevents viewings.
Also chains collapse all the time - what's your plan B if it doesn't work out? Can you move into a shared house/be a lodger/stay with relatives for a short while?
Unfortunately I don't really have a Plan B. I'm a self employed cake maker and make cakes in my domestic kitchen (5* hygiene rating before someone derails the thread there...). Needing the kitchen to myself, and for it to be kept very clean, doesn't exactly lend itself to situations involving being a lodger or in a houseshare.
I don't have family in the area - I moved up north for uni and never moved back - and need to be close to my customers for obvious reasons. You can't post a wedding cake! Regardless, DM wouldn't have space for me and DF has a kitchen that would require napalm total renovation before the EHO would be happy with it.
Only plan B I have is to drag out the eviction process as long as possible tbh, via the courts. It will buy me a few months, but not more. After that I'll be homeless and jobless.
When I inherited from my grandmother, the money from the house sale didnt come to me straight away. So the house sold, and there was still a load of admin to do before the money was distributed.
Oh crikey 
Could you go into a little more detail about the admin that was required after the house was sold? Had probate been granted before it was sold?